Here's a crazy thesis that nobody holds:
(1) If S knows that p, then S is permitted to assert that p.
There are boring counterexamples to (1). For instance, there are cases in which I am morally forbidden from asserting things that I know. This, of course, shows nothing interesting about the relationship between knowledge and assertion. So to capture the content of the interesting claim in the neighborhood, we move from (1) to something subtler and less crazy; maybe something like one of these:
(2) If S knows that p, then no epistemic shortcoming in S can explain why asserting that p would be inappropriate.
(3) If S knows that p, then S is in a strong enough epistemic position to assert that p.
(4) If S knows that p, then improving S's epistemic position won't put S in a position to assert p.
Maybe (2)-(4) are equivalent to one another; I'm not sure. They do a better job than (1) does at rendering certain kinds of cases of knowledge without assertability irrelevant. But they, too, are subject to conclusive refutation in a way orthogonal to the knowledge norm of assertion. My argument against (1) concerned cases in which I'm morally prohibited from asserting something, even though I know it -- maybe telling the Nazis where the Jews are hiding or something like that. But there are also cases in which I'm morally prohibited from asserting something, even though I know it, where a failure in my epistemic position plays a role in explaining the prohibition. There are boring cases like this. For instance, suppose that I've made a promise to assert that p only if I know that q. (Maybe promising by itself is insufficient for the relevant duties; stipulate that we're talking about a promise that carries serious moral weight.) Now suppose I know p, but don't know q. It's impermissible to assert that p, even though I know p -- and even though a deficiency in my epistemic position plays a role in explaining why p is unassertable. These cases, too, show that (2)-(4) don't get to the heart of the matter.
And it won't help to relativize the claims about strength of position to p, either. That is, these attempts at the knowledge norm face the same problem:
(5) If S knows that p, then S is in a strong enough epistemic position with respect to p to assert that p.
(6) If S knows that p, then improving S's epistemic position with respect to p won't put S in a position to assert p.
(7) If S knows that p, then p is, for S, warranted enough to justify S in asserting that p.
For take the special case of the sort of example given above where q is a proposition about S's epistemic position with respect to p. For instance, suppose I've promised not to assert p unless this condition is met: I know that I know that I'm absolutely certain that p. Suppose also that I know p, but don't know that I know that I'm absolutely certain that p. Under the circumstances, p is unassertable, because my epistemic position with respect to p isn't strong enough. But that doesn't show us anything interesting about the knowledge norm of assertion, if the latter is meant to be understood as showing something interesting distinctively about assertion.
What the knowledge norm of assertion suggests is that there's a special way that assertions can fail qua assertions. It says that if S knows p, then an assertion that p doesn't fail in this particular way. It doesn't provide any sufficient conditions for not failing in some other way, even when you build in all of these conditions about S's epistemic position.
Therefore, none of (1)-(7) are closely related to:
(Norm) Knowledge is the constitutive norm of assertion.
This post has gone on long enough for now, but I'll close by just asserting that many attempts to argue against the knowledge norm of assertion really look like arguments against some or all of (1)-(7); this is a mistake. I've argued that (1)-(7) are definitely false, for boring reasons that don't have anything to do with assertion in particular. If all you have is a case along with intuitions about what is known and what is unassertable, and why it's unassertable or under what circumstances that unassertability would be rectified, then you don't have anything strong enough to speak directly to (Norm). To evaluate (Norm) via the method of cases, you'd need to have intuitions about whether the assertion suffers from a particular kind of failure qua assertion. These, I think, we rarely have at any kind of pretheoretic level.